For the spin 1/2 fermions considered in question 3, prove the ``Hubbard Stratonovich'' identity

$$e^{-a (n_{\uparrow}-1/2)(n_{\downarrow} - 1/2)} = \frac{1}{2... ...igma = \pm 1} e^{\lambda\sigma (n_{\uparrow} -n_{\downarrow})}$$

where $\mbox{cosh} \lambda = e^{\frac{a}{2}}$ (Hint: consider the matrix representation of the operator on the left hand side in the four dimensional Fock space of the up and down electrons.) We shall later see how, considering $a \sim \delta t$ to be an infinitesimal time interval, we can use this relation to treat interacting spins as spins moving in a fluctuating Ising magnetic field.